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In my last imaging session with the spectroheliograph, I observed a strong filament in both the H alpha and Ca K images. Here are two images with the filament at, roughly, the solar equator towards the E limb chosen to be the one for study.

Montage_Finals_Ha-CaK lvls siz.png (501.82 KiB) Viewed 1177 times

An interesting aspect of the H alpha filament was its persistence over a large detuning from line centre. This can be appreciated in the spectral sweep animation presented here:

I thought the broad absorption (in wavelength span) from the H alpha filament might be due to a high temperature and corresponding large thermal (Doppler) broadening. Researching the subject a little, I came across the paper by Park et al.: “Temperature of Solar Prominences Obtained with the Fast Imaging Solar Spectrograph on the 1.6m New Solar Telescope at the Big Bear Solar Observatory” pgs105-116 in “Initial Results from the Fast Imaging Solar Spectrograph (FISS)” (2015) ed. Jongchul Chae, Springer International Publishing. These authors measured prominence temperatures and I decided to apply a similar analysis to the filament.

The analysis requires some elementary radiation transport theory and a measurement of the spectral profile for absorption by the filament from two atomic species, in the present case H and Ca(II). The starting point makes use of equation (2-19) in Foukal’s “Solar Astrophysics”, and reduces to the equation shown in the following figure.

The equation relates the transmitted intensity to the incident intensity on an absorbing medium (the filament) in terms of the optical depth / thickness and source function, at a given wavelength lambda. Note added 08Nov2018: I've since learned that this approach was introduced by J.M. Beckers (1964, PhD thesis , Univ. Utrecht ) as the "cloud model" and is discussed in detail in the article by Jongchul Chae, Astrophysical Journal, v780, p109 (11pp), 2014. The critical underlying assumption of validity in the cloud model, used in the present analysis, is that the source function is constant in position and independent of wavelength.

Foukal and others define and explain the significance of the optical depth / thickness and the source function. In fact, here are a couple of instructive videos:

Park et al. give the lambda dependence of the optical depth in terms of a Doppler broadened profile (Gaussian) as presented in the next figure.

filament spectral analysis fig2 siz.png (28.44 KiB) Viewed 1113 times

In addition to the temperature,T, there is a “non thermal” velocity component contributing to the Doppler width which, according to Park et al. “arises from all kinds of unresolved motion along the line of sight, including random motions of fine-structure threads”. This is something I'm not too familiar with so, time for more research. Note added 08Nov2018: This non thermal velocity component is also referred to as "microturbulence". It is described, for example, in the lecture notes of J.B.Tatum (University of Victoria, Canada):

The measurement of filament spectra arising from two different types of absorbing atoms (H and Ca) allows one to determine both the temperature and the non thermal velocity. The exercise involved fitting the spectral profile of a ratio of intensities: filament spectral intensity to underlying chromospheric spectral intensity. The model equation used is shown in the figure below has, as fitting parameters, the optical depth, the absorbing linewidth, the line centre wavelength and the source function. The fitting parameters were determined for both the H and Ca absorption spectra. As shown in the figure, a correction was applied to account for the instrumental spectral profile of the spectroheliograph.

Once the spectral linewidths are determined, by the fitting, for both H and Ca spectra, the two unknowns which determine the linewidth (temperature and non-thermal velocity) can be calculated. The assumption is that the H and Ca filaments are in thermal equilibrium (same temperature) and have the same non thermal velocities. The figure below shows the steps used to calculate the temperature, T. Once T is known, the non thermal velocity can be determined from either the H or Ca linewidth.

filament spectral analysis fig3 siz.png (24.44 KiB) Viewed 1113 times

The next figure shows the intensity ratio spectra and the location in the filament and adjacent portion of the chromosphere where the measurements were made. The H filament is shown, but, of course, the same measurement locations were used for both H and Ca filaments. The yellow squares in the surface image indicate the area over which an average spectrum was measured. For the spectrum plots, squares represent the actual measured intensities while curves represent the results of applying the fitting model described above. Plotting the spectra in terms of wavelength detuning from (chromospheric) line centre allowed me to superimpose the H and Ca results on to the same plot. Note that there’s an interesting “hump” near the centre of the H alpha filament absorption line. Park et al. noticed a similar result in some of their prominence emission lines (a dip near emission centre was observed) for large optical depth. This type of feature is called a “central reversal” (for both absorption and emission) and its absence in the model spectrum is due to the breakdown of a simplifying assumption used, namely that “the source function is constant along the line of sight”.

Fits to the spectral line profiles returned the values of four parameters: optical depth, line centre wavelength, linewidth and source function. In both H and Ca cases the line centre wavelength of the filament absorption was close to zero, indicating the filament was relatively stationary with respect to the underlying chromosphere. The results for optical depth, line centre wavelength and linewidth are shown in the next figure. The numerical value of the source function isn’t particularly relevant to the discussion here.

filament spectral analysis fig5 siz.png (32.71 KiB) Viewed 1113 times

Values obtained for the temperature (8930 K) and the non thermal velocity (10.2 km/s) are consistent with those measured by Park et al. in prominences. Park et al. were able to work with very high angular resolution and, consequently, could probe many separate regions in a given prominence. They showed a distribution of temperatures and non thermal velocities. My result is averaged over a fairly large region of the filament but falls well within their distribution.

So, what have I learned from this exercise?

First, the absorption line profile can have a relatively complicated shape but it can be well modelled by some simple radiation transport theory applied to the filament as an optically thick absorber. An optical depth of 1.0 usually serves as the dividing line between optically thick (> 1.0) and optically thin (< 1.0) absorbing media. The filament is clearly very optically thick in H and much less so in Ca. Hence the filament shows up more easily (with greater contrast) in H alpha images than in Ca images.

Second, the persistence of the filament in H alpha over a wide range of wavelengths around line centre (ie. the broad filament absorption lineshape) has little to do with temperature but a lot to do with optical depth. The filament is so highly absorbing near H alpha that light incident upon it (from below) is well attenuated even for wavelengths relatively far from line centre.

Third, an amateur spectroheliograph is capable of producing interesting solar images as well as some real scientific analysis of solar features!

Marvelous! Reading this almost makes me want to seek early retirement so I can pursue small science projects like this, too! Thanks also for the concise summary of the theoretical background, very interesting!

Frank: As a fairly new retiree I have to admit that I love doing these studies. I was always a science nerd, building questionable devices and frightening my parents when I was a kid. It’s nice to return to science as a hobby in my old age!